Constructions of irredundant orthogonal arrays

Abstract

An \begin{document}$ N \times k $\end{document} array \begin{document}$ A $\end{document} with entries from \begin{document}$ v $\end{document}-set \begin{document}$ \mathcal{V} $\end{document} is said to be an orthogonal array with \begin{document}$ v $\end{document} levels, strength \begin{document}$ t $\end{document} and index \begin{document}$ \lambda $\end{document}, denoted by OA\begin{document}$ (N,k,v,t) $\end{document}, if every \begin{document}$ N\times t $\end{document} sub-array of \begin{document}$ A $\end{document} contains each \begin{document}$ t $\end{document}-tuple based on \begin{document}$ \mathcal{V} $\end{document} exactly \begin{document}$ \lambda $\end{document} times as a row. An OA\begin{document}$ (N,k,v,t) $\end{document} is called irredundant, denoted by IrOA\begin{document}$ (N,k,v,t) $\end{document}, if in any \begin{document}$ N\times (k-t ) $\end{document} sub-array, all of its rows are different. Goyeneche and \begin{document}$ \dot{Z} $\end{document}yczkowski firstly introduced the definition of an IrOA and showed that an IrOA\begin{document}$ (N,k,v,t) $\end{document} corresponds to a \begin{document}$ t $\end{document}-uniform state of \begin{document}$ k $\end{document} subsystems with local dimension \begin{document}$ v $\end{document} (Physical Review A. 90 (2014), 022316). In this paper, we present some new constructions of irredundant orthogonal arrays by using difference matrices and some special matrices over finite fields, respectively, as a consequence, many infinite families of irredundant orthogonal arrays are obtained. Furthermore, several infinite classes of \begin{document}$ t $\end{document}-uniform states arise from these irredundant orthogonal arrays.